    Graph polynomials have been traditionally used to study properties of graphs, including adjacency (characteristic polynomial), coloring (Birkhoff's chromatic polynomial [1]}), Euler tours, rank polynomial [2]} and more. Apart from the standard graph polynomials, there is the Ihara zeta function [3]} which is useful in the study of free groups, spectral graph theory and symbolic dynamics. The famous Tutte polynomial [4]}, [5]}, [6]} encodes information about connectivity of induced subgraphs. It is equivalent to the Whitney rank generating function, related to the Jones polynomial (in knot theory), as well as connected to several computational problems in theoretical computer science. Historically, there have been connections between works in physics and graph polynomials. As as example, after Potts work [7]} on partition function of certain models in statistical mechanics in 1952, Fortuin and Kasteleyn [8]} found connections between Tutte polynomial and their work on random cluster model, a generalisation of the Potts model. In their work in 2010, Chang and Shrock [9]} had defined a polynomial \(Ph(G, q, w)\) , which generalizes the chromatic polynomial \(P(G,q)\) . Their aim was to study the statistical mechanics of the Potts antiferromagnet in a magnetic field as well as use these weighted graph coloring polynomials to solve problems that have physical applications. A recent study [10]} of the entanglement entropy of a topologically ordered state of quantum matter [11]}, [12]}, [13]} sheds light on the connection between multipartite quantum information measure and questions in graph theory.
